Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"NCG learning seminar: Isothermal Coordinates 1"

Time: 14:30

Room: MC 107

In this year's NCG learning seminar we plan to look at different topics of current interest to our group here at Western. We shall have regular weekly meetings on Wednesday afternoons. Exact time and place has yet to be fixed and will be announced after scheduling meeting next week. The first couple of lectures will be devoted to isothermal coordinates. Isothermal coordinates play an important role in conformal geometry of surfaces (both commutative and noncommutative), and in uniformization of Riemann surfaces. They also appear naturally in foliation theory (Godbillon-Vey invariant) and in the definition of entropy in the second law of thermodynamics. I shall first sketch a proof of their existence for general (analytic) Riemannian metrics and then discuss their relevance to other fields.

Algebra Seminar

Speaker: Piotr Aleksander Maciak (EPFL)

"Cancelled due to illness"

Time: 14:30

Room: MC 107

Geometry and Topology

Speaker: Christian Haesemeyer (UCLA)

"On the K-theory of toric varieties in characteristic p"

Time: 15:30

Room: MC 108

We will explain our recent joint work with G. Cortinas, M. Walker and C. Weibel concerning properties of the algebraic $K$-theory of toric varieties in positive characteristic. The results are proved using trace methods and a variant of the cyclic nerve construction that provides a homotopy theoretical model of the so-called Danilov sheaves of differentials. Most of the technical work happens completely within the world of monoid schemes (which are a particular manifestation of what goes by the name of "schemes over the field with one element").

In two subsequent less formal talks, I will endeavour to explain the theory of presheaves of spectra on monoid schemes, and the way topological cyclic homology is used in the constructions and proofs.

Algebra Seminar

Speaker: Guillermo Arturo Mantilla Soler (EPFL)

"The spinor genus of the integral trace and weak arithmetic equivalence"

Time: 15:30

Room: MC 107

In this talk I'll explain my recent results on the spinor genus of the integral trace form of a number field. I'll show how from them one can decide in terms of finitely many ramification invariants, and under some restrictions, whether or not a pair of number fields have isometric integral trace forms. Inspired by the work of R. Perlis on number fields with the same zeta function, I'll define the notion of weak arithmetic equivalence, and I'll show that under certain hypothesis this equivalence determines the local root numbers of the number field, and the isometry class of the integral trace form.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"NCG Learning Seminar: Introduction to Cyclic Cohomology I"

Time: 14:30

Room: MC 107

I will start by defining the Hochschild cohomology of an algebra $A$ with coefficients in an $A$-bimodule $M$. Some examples will be given and special attention will be paid to the case $M=A^*$, the linear dual of $A$. I will then show that Hochschild cohomology is a derived functor, and will use this technique to compute the Hochschild cohomology of noncommutative tori. Following this, I will introduce the cyclic cohomology of an algebra $A$, and will derive Connes' long exact sequence, which provides a powerful tool for computing cyclic cohomology.

Geometry and Topology

Speaker: Christian Haesemeyer (UCLA)

"On the K-theory of toric varieties in characteristic p, II"

Time: 15:30

Room: MC 108

Colloquium

Speaker: Paul Goerss (Northwestern)

"Algebraic Geometry and Large Scale Phenomena in the Homotopy Groups of Spheres"

Time: 15:30

Room: MC 108

A basic problem in algebraic topology is to write down all homotopy classes of maps between spheres. This problem, simple to state, is impossible to solve -- we don't even have a working guess. However, we've gotten very good at using some very specialized bits of algebraic geometry (the theory of abelian varieties and $p$-divisible groups, to be be precise) to get some hold on large scale patterns in these groups. After talking about how this connection works, I'll review some of the basic examples, going back even into the 1960s, then talk about the work of Hopkins-Miller-Behrens in the 2000s that uncovering some remarkable patterns using modular forms. This is only a start, and I'll end with some current vistas.

Algebra Seminar

Speaker: Winfried Bruns (Osnabr$\mathrm{\ddot{u}}$ck)

"Cancelled due to illness"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"iNCG"

Time: 14:30

Room: MC 107

To kick-start this year's noncommutative geometry (NCG) seminar, I shall give a quick survey of some major recent results. They are mostly related to analytic/geometric aspects of spectral triples and their applications.

Analysis Seminar

Speaker: Alexey Popov (University of Waterloo)

"Almost-invariant subspaces of operators and operator algebras"

Time: 14:30

Room: MC 108

In this talk, we will show that any bounded operator on a separable, reflexive, infinite-dimensional Banach space admits a rank-one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.

We will also show that if a (norm-closed) algebra A of operators on the Hilbert space has a non-trivial common almost-invariant subspace X (i.e., every member T of A can be perturbed by a finite-rank operator F_T so that X is invariant for T-F_T), then A admits a genuine non-trivial invariant subspace. Time permitting, we will talk about operators having many almost-invariant subspaces. Our principal result here is: if every projection from a masa produces an almost-invariant subspace for the operator T, then T=D+F where D is in the masa and F is finite-rank. This is a finite-rank version of a result of Johnson and Parrott from 1972.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Mathematics and Physics of the Quantum Hall Effect"

Time: 14:30

Room: MC 107

Klaus von Klitzing was awarded a Nobel prize in physics in 1985 for his discovery of the quantum Hall effect in 1980. In this talk I shall first give a quick introduction to the physics of the quantum Hall effect. I shall then explain how quantization of the Hall conductivity at low temperatures can be understood using tools from noncommutative geometry as pioneered by Jean Bellisard. Connes' index theorem relating K-theory and cyclic cohomology plays an important role here. (Note: This Thursday von Klitzing will give a talk at 5:30 PM in Talbot College.)

Homotopy Theory

Speaker: Daniel Schaeppi (Western)

"Localizations and completions in topology: An introduction"

Time: 14:30

Room: MC 108

Geometry and Topology

Speaker: Christian Haesemeyer (UCLA)

"On the K-theory of toric varieties in characteristic p, III"

Time: 15:30

Room: MC 108

Algebra Seminar

Speaker: Martin Frankland (Western)

"L-complete modules"

Time: 14:40

Room: MC 108

The category of L-complete modules is an interesting abelian subcategory of R-modules (for a nice ring R) which appears notably in algebraic topology. We will describe different interpretations of this category and some of its features. We will then discuss how completeness interacts with additional structure, such as algebras and lambda-algebras over R.

Noncommutative Geometry

Speaker: Nigel Higson (Penn State University)

"On the analytic approach to the quantization commutes with reduction problem"

Time: 14:30

Room: MC 107

The main goal of this lecture is to illustrate in one extended example some basic topological techniques in the C*-algebraic approach to index theory. The quantization commutes with reduction phenomenon (which I shall explain from scratch in the talk) was first explored by Guillemin and Sternberg within the context of Kahler geometry. A great deal has been achieved since then, but I want to return to the original complex-geometric context and examine there a remarkable Dirac operator approach developed by Tian and Zhang. This was originally framed within the context of symplectic geometry, but it simplifies considerably in the Kahler case, especially from the C*-algebra point of view.

Geometry and Topology

Speaker: Shintaro Kuroki (Univ. of Toronto/Osaka City Univ.)

"Root systems of torus graphs and characterization of extended actions of torus manifolds"

Time: 15:30

Room: MC 108

Torus manifold is a compact oriented $2n$-dimensional $T^n$-manifold with fixed points. We can define a labelled graph from the given torus manifold as follows: vertices are fixed points; edges are invariant $2$-dim sphere; edges are labelled by tangential representations around fixed points. This labelled graph is called a torus graph (this may be regarded as the generalization of special class of GKM graph). It is known that the equivariant cohomology of torus manifold can be computed by using combinatorial data of torus graphs. In this talk, we study when torus actions of torus manifolds can be induced from non-abelian compact connected Lie group (i.e., when torus actions can be extended to non-abelian group actions). To do this, we introduce root systems of torus graphs. By using this root system, we characterize what kind of compact connected non-abelian Lie group (whose maximal torus is $T^n$) acts on torus manifold. This is a joint work with Mikiya Masuda.

Analysis Seminar

Speaker: Raphael Clouatre (University of Waterloo)

"Classification of $C_{0}$ contractions"

Time: 14:30

Room: MC 108

We study the classification of Hilbert space contractions belonging to the class $C_{0}$ via their relation with their Jordan models.This classification is carried out in two different settings: up to similarity and up to unitary equivalence, both of which are stronger than the usual quasisimilarity relation that is known to always hold between a $C_{0}$ contraction and its model.

We obtain positive results under a variety of assumptions, ranging from function theoretic (the Vasyunin approach using the Carleson condition for sequences of inner functions) to operator algebraic (the Arveson approach using boundary representations of operator algebras).

Colloquium

Speaker: Nigel Higson (Penn State University)

"An incomplete introduction to noncommutative geometry"

Time: 15:30

Room: MC 108

The general aim of Alain Connes' noncommutative geometry is to develop selected geometric ideas using techniques from Hilbert space theory, with the goal of then transporting these concepts to new and unfamiliar contexts. But in this lecture I'll stay mostly with the familiar. I'll start with Hermann Weyl's theorem on eigenvalue asymptotics, examine it from the noncommutative-geometric point of view, and explain how a tentative idea of what a noncommutative geometric space might actually be starts to emerge in the process.

Noncommutative Geometry

Speaker: Nigel Higson (Penn State University)

"A geometric perspective on induction and restriction in tempered representation theory"

Time: 14:30

Room: MC 107

This talk is about the decomposition of the regular representation of a group like SL(2,R) into its irreducible constituents. The decomposition has both continuous and discrete parts, and more specifically my talk is about the continuous part, which arises through so-called parabolic induction. I'll describe a Hilbert bimodule construction, due to Pierre Clare, that places parabolic induction in a noncommutative-geometric, or C*-algebraic, context. It is also possible to construct an opposite "parabolic restriction" bimodule, although this is not at all trivial. The question of whether the induction and restriction bimodules are adjoint to one another has some interesting geometric aspects. This will be the main focus of the talk.

Homotopy Theory

Speaker: Marcy Robertson (Western)

"Localization of spaces with respect to homology"

Time: 14:30

Room: MC 108

Algebra Seminar

Speaker: Geoff Wild (Western)

"The logic of cooperative breeding"

Time: 14:30

Room: MC 108

In cooperatively breeding species, individuals help to raise offspring that are not their own. We use two population-genetic models to study the advantage of this kind of helpful behaviour in social groups with high reproductive skew. Our first model does not allow for competition among relatives to occur, but our second model does. Specifically, our second model assumes a competitive hierarchy among nestmates, with non-breeding helpers ranked higher than their newborn siblings. For each model we obtain an expression for the change in inclusive fitness experienced by a helpful individual in a selfish population. The prediction suggested by each expression is confirmed with computer simulation. When model predictions are compared to one another, we find that helping emerges under a broader range of conditions in the second model. Although competition among kin occurs in our second model, we conclude that the life-history features associated with this competition also act to promote the evolutionary transition from solitary to cooperative breeding.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"iNCG 2"

Time: 14:30

Room: MC 107

I shall give a quick survey of some major recent results in noncommutative geometry. They are mostly related to analytic/geometric aspects of spectral triples and their applications.

Geometry and Topology

Speaker: Hugo Bacard (UWO)

"Homotopy theory of co-Segal categories"

Time: 15:30

Room: MC 108

Given a monoidal model category $M$, we introduce a theory of co-Segal $M$-categories which are homotopy enriched categories over $M$. Examples of such categories emerge naturally when we consider homotopy transfers of algebraic structure. In this talk I will present the theory along with some examples and then will focus on the homotopy theory of these structures. Different model structures for co-Segal categories exist and I will talk about the canonical model structure, which is somehow the correct one.

Analysis Seminar

Speaker: Stamatis Pouliasis (Univ. Laval)

"On the asymptotic behavior of the capacity of certain condensers"

Time: 14:30

Room: MC 108

First we shall present some basic facts about condenser capacity, Green functions and their relation with complex analysis. Then we will examine the asymptotic behavior of the capacity of the inverse image of a condenser under exponential Blaschke products and universal covering maps.

Homotopy Theory

Speaker: Marcy Robertson (Western)

"Localization of spaces with respect to homology, part 2"

Time: 14:30

Room: MC 108

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Noncommutative Chern-Simons Theory"

Time: 14:30

Room: MC 107

After introducing classical Chern-Simons gauge theory, I will go over the noncommutative version of the theory and also the recent results on computing the action explicitly on some non-commutative spaces.

Colloquium

Speaker: Rasul Shafikov (Western)

"Lagrangian inclusions and holomorphic discs"

Time: 15:30

Room: MC 108

Gromov's theorem on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of $\mathbb C^n$ has had a deep impact on symplectic topology and complex analysis. I will discuss generalizations of this result to singular submanifolds.

Algebra Seminar

Speaker: Stefan Gille (Alberta)

"Permutation modules and motives of geometrically rational surfaces"

Time: 14:30

Room: MC 108

I will explain how permutation modules can be used to compute the motive of a geometrically rational surface. As a by-product one gets that a geometrically split motive with rational coefficients is always 0-dimensional.